On the asymptotic stability of bound states in 2D cubic Schroedinger equation

Abstract

We consider the cubic nonlinear Schrödinger equation in two space dimensions with an attractive potential. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small, localized in space initial data, converge to the set of bound states. Therefore, the center manifold in this problem is a global attractor. The proof hinges on dispersive estimates that we obtain for the non-autonomous, non-Hamiltonian, linearized dynamics around the bound states.

27 pages; references added

References in corpus (6)

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• A Rigorous Derivation of the Gross-Pitaevskii Energy Functional for a Two-Dimensional Bose Gas
• Selection of the ground state for nonlinear Schroedinger equations
• Dispersive estimates for Schroedinger operators in dimension two
• Stable manifolds for an orbitally unstable NLS
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